CC BY
5
ISSN 2304-0122 Ufa Mathematical Journal. Vol. 16. No 3 (2024). P. 40-53.
doi: 10.13108/2024-16-3-40
INTERPOLATION SETS IN SPACES OF FUNCTIONS OF FINITE ORDER IN HALF PLANE
M.V. KABANKO, K.G. MALYUTIN
Abstract. We consider free interpolation problems, the study of which was initiated by A.F. Leontiev. We obtain new criterions for the interpolation property of sets in the space of analytic in the upper half-plane functions of finite order. We provide examples of interpolation sets in the space of analytic in the upper half-plane functions of finite order. These examples are similar to interpolation sets in the space of analytic and bounded in the upper half-plane functions. In particular, we provide examples of sets satisfying the Newman condition and uniform Frostman condition.
Keywords: free interpolation, half-plane, finite order, interpolation set.
Mathematics Subject Classification: 30E05, 30D15
1. Introduction
1.1. Notation and terminology. If an inequality (identity) holds for all sufficiently large values of a variable, then it is called asymptotic inequality (identity). Bv K, M,... ,e,5,... we denote positive constants, which can change in arguing. For instance, we can use the phrase «if f (r) < 3M, then f (r) < M».
By N := {1, 2,... } we denote the set of (natural) numbers, C is the complex plane with the real axis R and positive semi-axis R+ := {x G R : x ^ 0} C+ := {z G C :Im z > 0} is the upper half-plane. One-point sets are written without braces if this causes no ambiguity. For instance, R := R U ±ro and R+ := R U +ro are respectively extended real axis and positive semi-axis with usual modulus | ■ | as for ^^d | ±ro| := +ro, ro is the infinity in the complex half-plane C+, that is, the sequence of points zn ^ ro as n ^ ro if lim |zra| = +ro, ni,n2 is
the set of integer numbers n : n\ ^ n ^ n2. The open circle of radius r centered at a point a is denoted by C(a,r),by B(a,r) = C(a, r) we denote a closed circle, G+ denotes the intersection of the set G with the half-plane C+, that is, G+ := G H C+, G is ^te closure of the set G.
Bv a+ we denote (|a| + a)/2, in particular, ln+ 0 := ^^ By [■ ] we denote the integer part of a number, A = {an}^=l C C+ is a sequence of points without repetitions with limiting points only on the real axis and at ro. Hereinafter, unless otherwise stated, we suppose rn = |ara|, 9n = argan, r = lz^ 9 = argz, where 0 ^ argz ^ n for z G C+ n+ (G) := n+(G) = Y1 sin9n,
a„eG
in particular, n+(R) := n+(C(0,R)).
M.V. Kabanko, K.G. Malyutin, Interpolation sets in spaces of functions of finite order in
half-plane.
(c) Kabanko M.V., Malyutin K.G. 2024.
The research of the second author is supported by the Russian Science Foundation, grant no. 24-21-00006, https://rscf.ru/project/24-21-00006/. Submitted February 3, 2024.
1.2. Definition of order of analytic in upper half^plane function. Let p (r E R+) be
the proximate order in the Valiron sense, lim p(r) = g > 0, We denote V(r) = rp(r\ Let f be
T—y^o
a holomorphie function in C+. The proximate order p is called semi-formal order of a function f if there exists a constant M > 0 (depending on f and independent of z) such that for all zE C+ the inequality
ln | f(z)l <MV(M)
holds and Levin condition is satisfied: there exist numbers q E (0,5 E (0,n/2) such that in each domain
D(R, q, 5) = {z : qR < |z| < R/q, 5 < argz < n —
there exist a point z, at which the inequality ln | f(z)l > — MV(|z|) holds.
The definition of semi-formal order of function is due to A.F. Grishin (see, for instance, [1]), We denote by [p, rc>)+ the space of functions, for which p is the semi-formal order,
1.3. Interpolation problem in space [p, The problems we consider in this work are free interpolation problems, the study of which was initiated by Leontiev [2] [4], A multiple interpolation problem in the space [p, q> 0, was solved in works [5], [6], The formulation of problem and theorem given below are its particular case, when the multiplicities of the interpolation nodes are equal to one. The case q = 0 (zero order) was considered in [7].
We give some notion and notation. We introduce the Nevanlinna canonical factor
' v(u — v)
Bg (u, v) = <
. for q = 0,
v(u — v)
B0(u, w)expiy^ — ( — — — || for qe N.
Let p ^e ^te proximate order in the Valiron sense, lim p(r) = Q > 0 Q = If a sequence A such that
n+(R) ^ KV(R) (1.1)
for some K > 0 (that is, it has a finite upper argument density) and q is a non-integer number, then the infinite product
E(z) := Ea(z) = n mz, On) h Bi(z,a™)
r„>\
converges uniformly on compact sets in C+,
E( ) A
The case of integer q ^ 1 is more complicated. In this case for the uniform convergence of the function E(z) on compact sets in C+ a finite upper argument density is not enough, we need some argument symmetry of the points an. For constructing the canonical product we add a factor without zero, the total measure of which is concentrated on the real axis (see [6]),
A
A
nj)(C(z,alzl) \ {an})
>D yjj^) :
+ - nD (
$+ (z ,a)
where an is the point in the support of the sequence A closest to the point z (if there are several such points, we choose an arbitrary of them). We let
/1 (.-, i) = *nef* +flZ,a\d:, 0=arg,
J a (a + sin (J)2
Definition 1.1. The sequence A = {an}X=i C C+, all limiting points of which belong to R U to, is called interpolating for the space [p, to)+ if for each, sequence of complex numbers bn, n E N, obeying the condition
ln+ | bnl
sup TTTH—h < ^
n V(lan|)
there exists a function F E [p, to)+ such, that
F(an) = bn, n E N. (1.2)
Hereafter we suppose that the condition |an| ^ 1 holds; this is a technical condition and it can be easily omitted. Let us formulate the versions of the theorem from [5], [6] for the case of simple interpolation.
Theorem 1.1. The following three statements are equivalent.
1) The sequence A is interpolating for the space [p, to)+.
2) If Q = lim p(r) is non-integer, then the canonical product of the sequence A satisfies the condition
sup ---— ln -:---— < TO. (1.3)
n V(|On|) lmanlE'(a,n)l
2') If Q ^ 1 is integer, then it follows from 1) that the condition (1.3) is satisfied for each, adjoint function E(z) of the sequence A. And vice versa, if (1.3) holds for at least one adjoint function E(z) of the sequence A, then 1) holds.
3) Condition (1.1) holds and for each 8 > 0
sup I+(z, i) < TO.
Theorem 1.2. The following two statements are equivalent.
1) The sequence A is an interpolating sequence for the space [p, to)+.
2) The condition (1.1) holds and for each 8 > 0
ln
an ak
<V ( rn), n = k, (1.4)
$+(a) ^a, ^ O (9 = argz), (1.5)
T +. . sin 9 sin d . $+(a) ^ -, 0 ^ -. (1.6)
z \ ) ln sin 0 ' ^ 2
a
1.4. Interpolation condition in space [p, to)+. Let a sequenee A = [an = rne%dn
belong to the upper half-plane A E C+ and there exist K > 0 such that conditions (1.1) are satisfied and
ak — ar,
n
rn/2<rk<3rn /2 k=n
ak — dr,
> exp [-KV(rn)]. (1.7)
A
[p, to) + (or, following [10], X+(p)-condition). The meaning of the condition (1.7) is that each
A A
the space [p, to)+.
Interpreting the half-plane C+ as a model of the plane in the Lobachevskii geometry, by a(z\, z2) we denote the non-Euclidean distance between aribtrarv points z^d z2 in the halfplane C+:
a( zi, Z2) =
1 + u
1 — u
u
Zi - Z2 Zi - Z2'
We can write the condition (1.7) as
JJ tan (2 a(ak,an)) ^ exp [-KV(rn)]
rn/2< rk<3r „/2 k=n
Here is another writing of the condition (1.7):
^T G(ak ) ^ KV(rn),
rn/2< rk <3r n/2 k=n
where
G(z, 0 = In is the Green function of the half-plane C+,
z a
-
2. Other conditions
In this section A = {an = rne%dn} still denotes a sequence in the upper half-plane, An
def
A \ {an} an G A.
1) (Sparse conditions.) The sequence A is called sparse (or p+-sparse) if there exists K > 0 such that the inequality
ak — ar,
ak — ar,
^ exp [-K(rn)]
holds.
It is clear that an interpolating for the space [p, to) + sequence A is necessarily p+-sparse, 2) (Newman condition.) We remunerate the sequence A so that
an+i — 1
ün+i + i
>
Qn 1
an + i
Let A satisfy the condition (1.1). Suppose that there exists c, 0 < c < 1, such that the A
Im an+i rl Im an
^ c
n+i
holds. Following [10], we say that the sequence A satisfies the eondition (N+(p)),
A
for order p (following [10], condition T+(p)) if it satisfies the condition (1.1) and there K > 0
sup V ,Ima\, ^ KV(rn).
ieR r„/2<rfc^3,n/2 lak -tl
3. Various reformulations of interpolation condition
Theorem 3.1. Let the sequence A = |ara = rne%dn belong the half-plane C+. The following statements are equivalent
1, The sequence A is an interpolating sequence for the space [p, to)+.
2,1, The condition (1,1) holds.
K > 0
rn (A)
def
E in ( i +
r„/2<rk<3r„ /2 k=n
Im ak Im a,, |ak - ara|2
« KV(rn)
holds.
3, The sequence A satisfies the condition ^+(p) and 3,1, The condition (1,1) is satisfied; K > 0
Sn(A)
def
E
rn/2<rk <3rn/2
i+« kv (r„)
|afc - ara|2
holds.
K > 0
sn(A) = ^ 'mak_ima2n « kv(rn)
rn/2<rk < 3rn / 2 a ^
holds.
Proof. We first of all mention two important simple identities
a
a — b
a — b
a
i
1 +
4 Im a Im b
| aa - | 2
4 Im a Im |a — 6|2 ,
a,b e C+,
a,b e C+, a = b.
Indeed,
aa
|a — 6|2 — |a —6|2 (a — b)(a — b) — (a — b)(a — b)
1 —-;-777:-1 —-
|a — b\'
|a — 6|2
1
= 1 +
| a |2 — ba — ab + |6|2 — | a |2 + ba + ab — |6|
|a — 6|2
(a — a)(b — b) 4ImaIm6
| a — | 2
1
| a — | 2
Similarly,
1
|a — b|2 — |a — 6|2
|a — b\2
(a — a)(b — b) 4ImaIm6
1 — --;-"^Vtt;—- = 1 +
|a — 6|2
| a — | 2
(3.1)
(3.2)
(3.3)
(3.4)
Let us prove the implication 1) 2). It follows from the identity (3.4) and condition (1.4) that
4Imafc Ima„\ 1 + —-— I « exp[ KV(rn)].
n
rn/2<rk <3rn/2 k=n
| ak — ara|2
2
2
2
2
2
Taking the logarithm, by this inequality we get (3,1),
We proceed to proving the implication 2) 3), It follows from (3,1) and (3,4) that
2 In
^ Tn(A) ^ KV (rn)
for n = k This implies the inequality
n- k
n- k
exp
K
- ^V (r n)
and this is why the sequence A satisfies the eondition rR+(p). Moreover, Sn(A) ^ rn(A), since for a,b E C+ the inequality
a-b < 1
a — b is true,
A
polation condition, then by the identity (3,3) and the elementary inequality — ln(1 — x) ^ x (0 ^ x < 1) we have
KV(rn) ^ - In
n
rn/2< rk <3r ri/2 k=n
ak - an
ak - an
£
In 1 -
rn/2< rk<3r n/2
for each point an E A. This implies (3,2),
3)
imply the inequality
Im ak Im ar Wk - ânl2
) >.. S.
rn /2< rk<3r n/2 k=n
Im ak Im an W - ânl2
1)
sup
£
k=i
Im ak Im an Wk - ânl2rek+i
< oo.
Indeed, the condition (1,1) implies the convergence of the series
Ki = £
sin dk
r k+i k=i 'k
<.
(3.5)
(3.6)
By (3.2) and (3.6) we obtain
£
k= n
Im ak Im an lâk - anl2rl+l
^ + ^ ^ 2 k+2Sn(A) + 4Ki sin 9n
rn/2< rk <3r n/2 lak-anl> rn/2 k= n
Since for k = n the corresponding summand in (3.5) is of the form 1/(4r'pl+1), we obtain (3.5).
We note that the condition (3.5) played a main role in works [5], [6] (see also [7]-[9]) for constructing a series, which solved the interpolation problem. The proof is complete. □
4. Union of interpolating sequences
It is easy to understand that the union of two interpolating sequences not necessarily possesses the interpolation property since the points of one set can closely approach the points of the other set. However, the following lemma holds.
Lemma 4.1. The .sequence A obeying the condition %+(p), which is equal to the union of several interpolating sequences, is an interpolating sequence for the space [p,
n
Proof. Let Ai, Ai, ..., Aq be interpolating sequences. Then the measures (j = 1, 2,... ,q) satisfy the conditions (1,5) and (1,6), The sum of these measures obviously satisfies these
conditions and hence, the measure p,A satisfies them as well, where A = (J Aj. If A is a
3 = 1
sequence satisfying the condition rR+(p), then it also satisfies the condition (1,4) of Theorem 1,2, Therefore, by Theorem 1.2 the sequence A is an interpolating sequence in the space, □
5. Relations between conditions Z+(p) and N+(p)
Lemma 5.1. A sequence A satisfying the condition N+ (p) is an interpolating sequence for the space [p,
Proof. The condition N+(p) and the elementary inequality
imply
(|a + z| — |a — z|)|6 + z| ^ gIma |6|2 |a + il(lb + z| — lb — z|) ^ Im& |a|2
| an+i + il — |an+i — i\ lan + z| — |an — z| — ^ c-
|«n+1 + A | «n + A
Since for all points a, b in the upper half-plane C+
a
a
we have
n
rn/2<rj <3rn/2 j=n
aj an
aj an
>
n
|a9
r„/2<rj <3r„/2 J j>n
|a — illb + z | — | a + illb — z| |a + z||6 + z| — |a — illb — z|'
| a j — ill an + z | — | aj + z || an — z |
n
a 7 + z||ara + z| — l aj — z|| ara — z| an — z ll aj + z l — l an + z || a,- — i l
r„/2<rj <3r„/2 J j<n
aj + z||ara + z| — a — z|| ara — z
n ■ n •
3>n j<n
(5.1)
If j > n, then
|aj + z| — |aj — i\ < \an + z| — |an — z|
| aj + z| | ^ +
and thus,
laj - z||an + A - |o,- + z||on - z| ^ (1 - cj-n)laj + z|(|on + z| - |an - z|). On the other hand,
K' + ¿|| an + z| - laj - z||an -A ^ (1 + cj-n)laj + Ad^n + - | ^ - z|).
Thus,
n * n
j>n j=1
1 — cJ
1 + Cj'
We then find
'n n = — £ 'nf—^ « 2 £
j>n j=1
By elementary calculations we obtain
j=i
1 + cJ
ln n » 2 £
j>n
—' 1 + cJ In c ■ ln(1 + c)'
(5.2)
j
3
2
If j < n, then
lan + il - lan - i\ n_, laj + il - Idj - il
-1-n-^ c -1-n-•
lan + tl l aj + il
Thus,
lan - z||aj + il - lan + illaj - il ^ (1 - cn j )lan + z|(|aj + il - laj - z|), la,n + illaj + il - laj - illa,n - il ^ (1 + cn~j)lan + il(laj + il - laj - z|).
Hence,
and
n»n
^ 1 _ cn-j
1 + Cn~j
j<n j=l
2
lnTT -r-Ti-7. (5.3)
11 ln c • ln(1 + c)
j<n x '
It follows from (5.2) and (5.3) that the sequence A obeys the eondition (X+(p)). □
Corollary 5.1. Each sequence, the limiting points of which are located on the real axis and at infinity, contains an interpolating for the space [p, subsequence.
Corollary 5.2. Let a sequence A satisfy the condition (X+(p)) and all its points are on the imaginary axis. Then the necessary condition for A being interpolating for the space [p, is
a
^ exp[KV(r-n)]
an+i
for -some K > 0, and the sufficient condition is
a 'n < c< 1.
an+1
Proof. We have already shown that the interpolation is possible if an tends to the boundary (in the present case to infinity) with an exponential rate. And vice versa, if the interpolation
K > 0
r t^ta/ ^ an+l an an+l an ^ an+l
exp[-KV(r,n)] ^ -—— =--— ^ -.
an+1 an an+1 + an an
□
6. Interpolation of sparse sequence obeying uniform Frostman condition
In this section A = {an = rne%dn} still denotes a sequence in the upper half-plane, An =f A \ {an} an e A.
Lemma 6.1. A p+-sparse sequence A satisfying the condition (T+(p)) is an interpolating sequence.
Proof. Let A0 be a finite subsequence of the sequence A. The function
Im ak
/«= £
ak — z\
ak eAo
is subharmonic in C+ and this is why
^ Im afc t(4\ ^ V^ Im ak
\a—v\ ^ max/(i) ^ sup z,
_ /2 lak -A ten ieR lak -t\
ak €Aq
Hence,
E
Im ak
|afc — a'n\
« KV(rn)
rn/2«rk«3rn/2
for each point an in the set A. At the same time
Im an
|ak — an\
< 1, au ,an E C+.
Therefore,
E
Im ak Im ar, |afc — ara|2
« KV(rn).
rn/2<rk<3rn/2 +
of the sequence A (Theorem 3,1), □
7. Sets close to interpolating ones
In what follows we are interesting in sets close to interpolating ones in the space [p, that is, to the sets obeying the Z+ (p)-condition. By Q(a, r) we denote the circle
tt(a, r) f z E C :
— a
— a
«
.
The set EMs K-translated with respect to the set E (E C C+) for proximate order p if there exists a mapping u of the set E onto the set E' such that w(£) E , exp [-KV(|£|)j) for all £ E E, In this case we say that the mapping u is K-tramlation of the set E for proximate
We being with proving some useful inequalities.
Lemma 7.1. Let a, b,c,d E C+,
a
u =
aa
— a — a'
w =
c — d a — d
1) V
then v ^ 5a, where
2) If
then w > 8al3, where
—
-« -,
b - a 4
u > 5,
a
—
-« -,
b - a 4
P
_ ln(ô/2) " in 5 .
a
--« -,
aa — 4
_ ln(5/4) " ln 5/2 .
u > 5,
Proof We apply the conformai mapping
w =
of the half-plane C+ onto the unit circle C(0,1) and similar lemma for the circle [10, Lm. 8] and this completes the proof, □
Lemma 7.2. Let a, b E C+, lal ^ 1. If
a
aa
^ e, 0 < e < 1,
then
1) the inequality holds
2) the inequality holds
Proof 1) We have This yields
This inequality implies
la -bl ^
arg -
2e Ima
T-7";
2e Ima 1 - £ .
aa 1 - ^ -.
a
1
2e Ima
a
1
^ -.
la - bl + 2Ima ^
la-bl
and hence, we obtain the inequality in Assertion 1),
2) The needed inequality follows from that in Assertion 1) since for |a| > 1 and la-bl ^ r < 1, we have | arg a — arg bl ^ r.
The proof is complete, □
Lemma 7.3. Let a sequence A = {an}^=1 C C+, an = rnet&", obeys the relation
^ sin On ^C\V(r) (7.1)
anec (0,r)
for some C1 > 0. Then for a given K > 0 there exists C2 = C2(K) > 0 such that each K-translated sequence A' = {aln}c^=1 C C+, a'n = r'netdn, for proximate order p(r) with respect to the sequence A satisfies the condition (7.1) with the constant C2.
Proof. Using Assertion 2 of Lemma 7.2, we obtain
^ sin dn ^ ^ sin On + ^ | sin 9'n — sin 9nl
a'nec(0,r) anec(0,2r) a'neC(0,r)
^ sin B,n + sin Bn exp(—C2V(r,n)), C2 > 0.
a„EC(0,2r) a„EC(0,2r)
The series sin Bn exp(—C2V(rn)) converges. Now inequalities (7.1) and (7.2) imply the
anec (0,2r)
statement of the lemma. □
Theorem 7.1. Let a sequence A obeys the T+(p)-condition. Then there exists a number K > 0
1) If an e A, ( e Q (an, exp [—KV(rn)]), then
ak — (
(7.2)
l^A(0l
n
rn/2< rk <3r n/2 ak eA„
ak
> exp [-KV(r-n)].
a
2) Each sequence A', which is K-translated for proximate order p with respect to A satisfies X+(p) -condition.
3) Each K-translation for proximate order p of the sequence A is a one-to-one correspondence.
4) If ak, an E A and ak = an, then
Q (ak, exp [-KV(rk)]) n Q (an, exp [-KV(r^)]) = 0.
K > 0 n E A
n
rn/2<rk<3rn/2//oo£An
ak - ar_
ak - dn
> exp [-KV(r^)].
This inequality yields
ak - an
^ exp [-KV(rn)].
We apply Assertion 1) in Lemma 7.1 with 5 = exp [-KV(rn)]. We let
a = ak, b = an, £,V E E, CE Q( 1, ¿exp[-KV(M]
1
)
and we get
n
|?|/2<|^|<3|?|/2//^eei
-
d-c
^ (^Vin(,) = 1 exp [-KV (Id].
C1 , K > 0
proximate order p(r), lim p(r) = p, such that the condition (1.7) holds for each ( E E
n
ICI/2<|TI<3|CI/2
1eEc
-
i-d
^ exp[-CiV(|<|].
This inequality yields
We let
-
7-d
^ -CiV(Id.
K = Ci, EE', 7] E 4q (7, exp [-KV (|7|]) , £ E 4q (C, exp [-KV (|<|)]).
Applying Assertion 2) of Lemma 7.1 with 5 = exp [-KV(|£|], we get
n
|?|/2<M<3|?|/2 veEs
-
^ (6)Hs/2)/m(W/4)/^/2) = 1 exp [-KV (|£|]
Assertions 3) and 4) follow from Assertion 2). Indeed, if
EE',
V E 4Q (7, exp [-KV(|7|]), e E 4Q «, exp [-KV(|<|])
then
-
> -CiV(|£| > 0,
=
□
8. Weakly regular sets
In [11], weakly regular sets in the upper half-plane [p(r), x)+ were studied. Their definition is as follows.
Definition 8.1. A sequence of points A = {an^^ in the upper half-plane C+ is called weakly regular sequence with, respect to the proximate order p (or WR+(p)-set) if one of the following conditions (C) or (C') is satisfied:
(C+) -condition
A
2) A nC(0, 2) = 0;
3) the condition holds
n+(C(0, r)) ^ KV(r), K> 0;
4) there exists a number d > 0 such that for all points an, ak e A such that lanl ^ lak| we have
1 an l ^ lakl + dim ak/ V(|ak|).
(C+) -condition
A
2) A nC (0, 2) = 0;
3) the condition holds
n+(C(0, r)) ^ KV(r), K> 0;
4) there exists a number d > 0 such, that the circles of radii
7/ • / \\1/2l l1 — P(lan
rn = d(sin(argan)) / lanl 2
an
The sets obeying (C+)-eondition played an important role in [12], [13],
By using the geometric criterion of the interpolation property of sequence, the following theorem was proved.
Theorem 8.1. Let a sequence A = {an}^=1, A e C+, be a WR+(p)-set. Then A is an interpolating sequence in the space [p, x)+.
9. Interpolation problem for compactly supported sequences
We recall that a sequence of complex numbers {bn}^=1 is called compactly supported if all terms in this sequence vanish starting from some one, that is, bn = 0 for n ^ n0 ^ 1,
Interpolation problems in spaces of entire functions take special place for compactly supported sequences and are related with the distribution of zeroes of entire functions. In particular, Bratishehev and Korobejnik [14] considered a problem on multiple interpolation in the space [p, X) of entire functions of finite order (include zero order) and normal type for compactly supported sequences. The obtained result can be formulated as follows.
Theorem 9.1 (Bratishehev, Korobejnik). The following three statements are equivalent.
1) The interpolation problem 1.2 in the space [p, x) is solvable for each, compactly supported sequence.
2) The set of interpolation nodes {an}n=n0to obeys the condition
n(r)
limsup < to,
r V'(r)
where n(r) is the number of points in the sequence {an}^=n0 located in the circle |z| < r.
3) The interpolation problem (1.2) in the space [p, to) is solvable for at least one non-zero compactly supported sequence.
We consider problem on simple interpolation for compactly supported sequences in the space [P, to)+-
Theorem 9.2. The following three statements are equivalent.
1) The interpolation problem (1.2) in the space [p, to)+ is solvable for each compactly supported sequence.
2) The condition (1,1) is satisfied.
3) The interpolation problem (1.2) in the space [p, to)+ is solvable for at least one non-zero compactly supported sequence.
Proof. The implication 1) ^ 3) is trivial. Let us prove the implication 3) ^ 2), Suppose that the interpolation problem (1,2) is solvable in the space [p, to)+ for a non-zero compactly supported sequence {bn}(^=v where bn = 0 for all n > n0 and bno = 0. Let a function F E [p, to]+ be such that
F(an) = bn, n =1, 2,... ,n0, F(an) = 0, n = n0 + 1,....
The zero set {zn} of the function contains the sequence {an}^=no and by Theorem 1.2 it obeys the condition (1.1). Since just finitely many terms in the sequence {an}^=i does not belong to the set {zn}, the sequence {an}c^=i also satisfies the condition (1.1). The proof of the implication 3) ^ 2) is complete.
We proceed to proving the implication 2) ^ 1). Suppose that the condition (1.1) is satisfied. Then the canonical product E(z) of the sequenee {an}^=i belongs to the space [p, to)+. Let {^n}<^=i be a compactly supported sequence. Since the series
F (z) = V _E (Z)b n_
( ) 2=i (z - an)E'(an)
contains just finitely many non-zero terms, the function F(z) belongs to the space [p, to)+. The function F(z) solves the interpolation problem (1.2). The proof of the implication 2) ^ 1) is complete. □
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Mikhail Vladimirovich Kabanko, Kursk State University, Radiseheva str. 33, 305000, Kursk, Russia E-mail: [email protected]
Konstantin Gennadievich Malyutin, Kursk State University, Radiseheva str. 33, 305000, Kursk, Russia E-mail: [email protected]
